The Hilbert–Smith conjecture for three-manifolds

John Pardon

9 April 2012; Revised 22 January 2013

Abstract

We show that every locally compact group which acts faithfully on a connectedthree-manifold is a Lie group. By known reductions, it suffices to show that there isno faithful action of Zp (the p-adic integers) on a connected three-manifold. If Zp actsfaithfully on M3, we find an interesting Zp-invariant open set U ⊆ M with H2(U) = Zand analyze the incompressible surfaces in U representing a generator of H2(U). Itturns out that there must be one such incompressible surface, say F , whose isotopyclass is fixed by Zp. An analysis of the resulting homomorphism Zp → MCG(F ) givesthe desired contradiction. The approach is local on M .

MSC 2010 Primary: 57S10, 57M60, 20F34, 57S05, 57N10MSC 2010 Secondary: 54H15, 55M35, 57S17Keywords: transformation groups, Hilbert–Smith conjecture, Hilbert’s Fifth Prob-

lem, three-manifolds, incompressible surfaces

1 Introduction

By a faithful action of a topological group G on a topological manifold M , we mean acontinuous injection G → Homeo(M) (where Homeo(M) has the compact-open topology).A well-known problem is to characterize the locally compact topological groups which canact faithfully on a manifold. In particular, there is the following conjecture, which is anatural generalization of Hilbert’s Fifth Problem.

Conjecture 1.1 (Hilbert–Smith). If a locally compact topological group G acts faithfully onsome connected n-manifold M , then G is a Lie group.

It is well-known (see, for example, Lee [17] or Tao [41]) that as a consequence of thesolution of Hilbert’s Fifth Problem (by Gleason [8, 9], Montgomery–Zippin [22, 23], as wellas further work by Yamabe [46, 47]), any counterexample G to Conjecture 1.1 must containan embedded copy of Zp (the p-adic integers). Thus it is equivalent to consider the followingconjecture.

Conjecture 1.2. There is no faithful action of Zp on any connected n-manifold M .

Conjecture 1.1 also admits the following reformulation, which we hope will help the readerbetter understand its flavor.

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Conjecture 1.3. Given a connected n-manifold M with metric d and open set U ⊆ M ,there exists ǫ > 0 such that the subset:

{

φ ∈ Homeo(M)∣

∣ d(x, φ(x)) < ǫ for all x ∈ U}

(1.1)

of Homeo(M) contains no nontrivial compact subgroup.

For any specific manifoldM , Conjectures 1.1–1.3 forM are equivalent. They also have thefollowing simple consequence for almost periodic homeomorphisms ofM (a homeomorphismis said to be almost periodic if and only if the subgroup of Homeo(M) it generates hascompact closure; see Gottschalk [10] for other equivalent definitions).

Conjecture 1.4. For every almost periodic homeomorphism f of a connected n-manifoldM , there exists r > 0 such that f r is in the image of some homomorphism R → Homeo(M).

Conjecture 1.1 is known in the cases n = 1, 2 (see Montgomery–Zippin [23, pp233,249]).By consideration of M × R, clearly Conjecture 1.1 in dimension n implies the same in alllower dimensions.

The most popular approach to the conjectures above is via Conjecture 1.2. Yang [48]showed that for any counterexample to Conjecture 1.2, the orbit space M/Zp must havecohomological dimension n + 2. Conjecture 1.2 has been established for various regularityclasses of actions, C2 actions by Bochner–Montgomery [4], C0,1 actions by Repovs–Scepin[32], C0, n

n+2+ǫ actions by Maleshich [18], quasiconformal actions by Martin [19], and uniformly

quasisymmetric actions on doubling Ahlfors regular compact metric measure manifolds withHausdorff dimension in [1, n+ 2) by Mj [20]. In the negative direction, it is known by workof Walsh [44, p282 Corollary 5.15.1] that there does exist a continuous decomposition ofany compact PL n-manifold into cantor sets of arbitrarily small diameter if n ≥ 3 (see alsoWilson [45, Theorem 3]). By work of Raymond–Williams [31], there are faithful actions of Zp

on n-dimensional compact metric spaces which achieve the cohomological dimension jumpof Yang [48] for every n ≥ 2.

In this paper, we establish the aforementioned conjectures for n = 3.

Theorem 1.5. There is no faithful action of Zp on any connected three-manifold M .

1.1 Rough outline of the proof of Theorem 1.5

We suppose the existence of a continuous injection Zp → Homeo(M) and derive a contra-diction.

Since pkZp∼= Zp, we may replace Zp with one of its subgroups pkZp for any large k ≥ 0.

The subgroups pkZp ⊆ Zp form a neighborhood base of the identity in Zp; hence by continuityof the action, as k → ∞ these subgroups converge to the identity map on M . By picking asuitable Euclidean chart of M and a suitably large k ≥ 0, we reduce to the case where M isan open subset of R3 and the action of Zp is very close to the identity.1

1There are two motivations for this reduction (which is valid in any dimension). First, recall that atopological group is NSS (“has no small subgroups”) iff there exists an open neighborhood of the identitywhich contains no nontrivial subgroup. Then a theorem of Yamabe [47, p364 Theorem 3] says that a locally

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The next step is to produce a compact connected Zp-invariant subset Z ⊆ M satisfyingthe following two properties:2

1. On a coarse scale, Z looks like a handlebody of genus two.

2. The action of Zp on H1(Z) is nontrivial.

The eventual contradiction will come by combining the first (coarse) property of Z with thesecond (fine) property of Z. Constructing such a set Z follows a natural strategy: we take theorbit of a closed handlebody of genus two and attach the orbit of a small loop connecting twopoints on the boundary. However, the construction requires checking certain connectednessproperties of a number of different orbit sets, and is currently the least transparent part ofthe proof.

Now we consider an open set U defined roughly as Nǫ(Z)\Z (only roughly, since we needto ensure that U is Zp-invariant and H2(U) = Z). The set of isotopy classes of incompressiblesurfaces in U representing a generator of H2(U) forms a lattice, and this lets us find anincompressible surface F in U which is fixed up to isotopy by Zp. We think of the surfaceF as a sort of “approximate boundary” of Z. Even though Zp does not act naturally onF itself, we do get a natural homomorphism Zp → MCG(F ) with finite image. The twoproperties of Z translate into the following two properties of the action of Zp on H1(F ):

1. There is a submodule of H1(F ) fixed by Zp on which the intersection form is:

(

0 1−1 0

)

⊕

(

0 1−1 0

)

(1.2)

2. The action of Zp on H1(F ) is nontrivial.

This means we have a cyclic subgroup Z/p ⊆ MCG(F ) such that H1(F )Z/p has a submodule

on which the intersection form is given by (1.2). The Nielsen classification of cyclic subgroupsof the mapping class group shows that this is impossible. This contradiction completes theproof of Theorem 1.5.

We conclude this introduction with a few additional remarks on the proof. First, theproof is a local argument on M , similar in that respect to the proof of Newman’s theorem[24]. Second, our proof works essentially verbatim with any pro-finite group in place of Zp

(though this is not particularly surprising). Finally, we remark that assuming the action ofZp on M is free (as is traditional in some approaches to Conjecture 1.2) does not produceany significant simplifications to the argument.

compact topological group is a Lie group iff it is NSS. Thus the relevant property of Zp which distinguishesit from a Lie group is the existence of the small subgroups pkZp. Second, recall Newman’s theorem [24]which implies that a compact Lie group acting nontrivially on a manifold must have large orbits. In essence,we are extending Newman’s theorem to the group Zp (however the proof will be quite different).

2The motivation to consider such a set is to attempt a dimension reduction argument. In other words,we would like to conclude that ∂Z is a closed surface with a faithful action of Zp, and therefore contradictsthe (known) case of Conjecture 1.2 with n = 2. This, of course, is not possible since Z could a priori havewild boundary. We will nevertheless be able to construct a closed surface F which serves as an “approximateboundary” of Z.

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1.2 Acknowledgements

We thank Ian Agol for suggesting Lemmas 2.14 and 2.19 concerning lattice properties ofincompressible surfaces. We also thank Mike Freedman and Steve Kerckhoff for helpfulconversations. We thank the referee for their comments and for giving this paper a veryclose reading.

The author was partially supported by a National Science Foundation Graduate ResearchFellowship under grant number DGE–1147470.

2 The lattice of incompressible surfaces

In this section, we study a natural partial order on the set of incompressible surfaces ina particularly nice class of three-manifolds; we call such three-manifolds “quasicylinders”(see Definition 2.4). For a quasicylinder M , we let S(M) denote the set of isotopy classesof incompressible surfaces in M which generate H2(M). The main result of this section(suggested by Ian Agol [1]) is that S(M) is a lattice (see Lemma 2.19) under its natural partialorder. Related ideas may be found in papers of Schultens [36] and Przytycki–Schultens [30]on the contractibility of the Kakimizu complex [14].3

Since we will ultimately use the results of this section to study groups of homeomorphisms,we need constructions which are functorial with respect to homeomorphisms. To studythe properties of these constructions, however, we use methods and results in PL/DIFFthree-manifold theory (for example, general position). Thus in this section we will alwaysstate explicitly which category (TOP/PL/DIFF) we are working in, since this will changefrequently (and there is no straightforward way of working just in a single category). Wewill, of course, need the key result that every topological three-manifold can be triangulated,as proved by Moise [21] and Bing [3] (see also Hamilton [12] for a modern proof based on themethods of Kirby–Siebenmann [16]; the PL structure is unique, but we do not need this).

By surface, we always mean a closed connected orientable surface. By isotopy (resp. PLisotopy) of surfaces, we always mean ambient isotopy through homeomorphisms (resp. PLhomeomorphisms).

Theorem 2.1 ([3, p62 Theorem 8]). Let M1,M2 be two PL three-manifolds where M2 hasa metric d. Let φ : M1 → M2 be a homeomorphism, and let f : M1 → R>0 be continuous.Then there exists a PL homeomorphism φ1 :M1 →M2 with d(φ(x), φ1(x)) ≤ f(x).

Lemma 2.2. Let M be a PL three-manifold, and let F be a bicollared surface in M . Thenthere exists an isotopy of M supported in an arbitrarily small neighborhood of F which mapsF to a PL surface.

Proof. Let φ : F × [−1, 1] → M be a bicollar, which we may assume is arbitrarily close toF = φ(F × {0}). Now pick a PL structure on F and apply Theorem 2.1 to φ|F×(0,1) and a

3The Kakimizu complex is a sort of “Z-equivariant order complex” of S(S3 \K), where S3 \K denotes

the infinite cyclic cover of the knot complement S3\K. Even though technically S3 \K is not a quasicylinderunder our definition, it is easy to give a modified definition (allowing manifolds with boundary) to which themethods of this section would apply.

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function f which decreases sufficiently rapidly near the ends of (0, 1). The resulting φ1 thenextends continuously to F × [0, 1] and agrees with φ on F ×{0, 1}. Now splice φ1|F×[0,1] andφ|F×[−1,0] together to get a bicollar φ2 : F × [−1, 1] → M which is PL on F × (0, 1). Nowusing the bicollar φ2 we can easily construct an isotopy of M which sends F = φ2(F × {0})to φ2(F × {1

2}), which is PL.

Lemma 2.3. Let M be an irreducible orientable TOP (resp. PL) three-manifold, and letF1, F2 be two bicollared (resp. PL) π1-injective surfaces in M . If F1, F2 are homotopic, thenthere is a compactly supported (resp. PL) isotopy of M which sends F1 to F2.

Proof. Waldhausen [43, p76 Corollary 5.5] proves this in the PL category if M is compactwith boundary. It is clear that this implies our PL statement, since the given homotopy willbe supported in a compact region of M , which is in turn contained in a compact irreduciblesubmanifold with boundary.

For the TOP category, first pick a PL structure on M , and use Lemma 2.2 to straightenF1, F2 by a compactly supported isotopy. Now use the PL version of this lemma.

Definition 2.4. A quasicylinder is an irreducible orientable three-manifold M with exactlytwo ends and H2(M) ∼= Z. For example, Σg × R is a quasicylinder for g ≥ 1.

Lemma 2.5. Let M be a quasicylinder. For an embedded surface F ⊆M , the following areequivalent:

1. F is nonzero in H2(M).2. F separates the two ends of M .3. F generates H2(M).

Proof. (1) =⇒ (2). A path from one end to the other gives a non-torsion class in H lf1 (M).

Thus its Poincare dual is a non-torsion class in H2(M), and thus defines a nonzero mapH2(M) → Z. Since F is nonzero in H2(M) ∼= Z, every such path must therefore intersect F .

(2) =⇒ (3). If F separates the two ends, then there is a path from one end to the otherwhich intersects F exactly once. Thus the Poincare dual of the class of this path in H lf

1 (M)evaluates to 1 on F ∈ H2(M). Thus F represents a primitive element of H2(M) ∼= Z, andthus generates it.

(3) =⇒ (1). Trivial.

Definition 2.6. For a TOP quasicylinder M , let STOP(M) be the set of bicollared π1-injective embedded surfaces in M generating H2(M), modulo homotopy.

Definition 2.7. For a PL quasicylinder M , let SPL(M) be the set of PL π1-injective em-bedded surfaces in M generating H2(M), modulo homotopy.

Remark 2.8. SPL(M) is always nonempty, since we can pick a PL embedded surface repre-senting a generator of H2(M) and then take some maximal compression thereof (as in theproof of Lemma 2.17 below), which will be π1-injective by the loop theorem.

Definition 2.9. A directed quasicylinder is a quasicylinder along with a labelling of the endswith ±. For an embedded surface F ⊆ M separating the two ends, let (M \ F )± denote thetwo connected components of M \ F (the subscripts corresponding to the labelling of theends). It is easy to see that (M \ F )± are both (directed) quasicylinders.

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Definition 2.10. LetM be a directed quasicylinder. For two embedded surfaces F1, F2 ⊆Mseparating the two ends of M , we say F1 ≤ F2 iff F1 ⊆ (M \ F2)−. Define a relation ≤ onSTOP(M) (resp. SPL(M)) by declaring that F1 ≤ F2 if and only if F1,F2 have embeddedrepresentatives F1, F2 with F1 ≤ F2.

Lemma 2.11. Let M be a PL directed quasicylinder. Then the natural map ψ : SPL(M) →STOP(M) is a bijection satisfying F1 ≤ F2 ⇐⇒ ψ(F1) ≤ ψ(F2).

Proof. The natural map (SPL(M),≤) → (STOP(M),≤) is clearly injective, and by Lemma2.2 it is surjective.

If F1,F2 have bicollared representatives F1, F2 with F1 ≤ F2, then by Lemma 2.2 theycan be straightened preserving the relation F1 ≤ F2. The other direction is obvious, sincePL surfaces have a bicollar.

Having established that SPL(M) and STOP(M) are naturally isomorphic, we henceforthuse the notation S(M) for both.

Lemma 2.12. Let M be a PL quasicylinder. Suppose F is a PL embedded surface in Mseparating the two ends of M , and suppose γ is a PL arc from one end of M to the otherwhich intersects F transversally in exactly one point. Denote by π1(M, γ) one of the groups{π1(M, p)}p∈γ (they are all naturally isomorphic given the path γ).4 Then for any surfaceG ⊆ M homotopic to F , there is a canonical map π1(G) → π1(M, γ) (defined up to innerautomorphism of the domain).

Proof. Assume that G intersects γ transversally. Let us call two intersections of γ withG equivalent iff there is a path between them on G which, when spliced with the pathbetween them on γ, becomes null-homotopic inM (this is an equivalence relation). Note thatduring a general position homotopy of G, the mod 2 cardinalities of the equivalence classesof intersections with γ remain the same. Thus since G is homotopic to F and #(F ∩ γ) = 1,there is a unique equivalence class of intersection G ∩ γ of odd cardinality. Picking any oneof these points as basepoint on G and on γ gives the same map π1(G) → π1(M, γ) up toinner automorphism of the domain. This map is clearly constant under homotopy of G.

Lemma 2.13. Let M be a PL quasicylinder. Let F be a PL π1-injective surface in Mseparating the two ends of M . Then any homotopy of F to itself induces the trivial elementof MCG(F ).

Proof. Pick a PL arc γ from one end of M to the other which intersects F transversallyexactly once. By Lemma 2.12, we get a canonical map π1(F ) → π1(M, γ) which is con-stant as we move F by homotopy. Since this map is injective, we know π1(F ) up to innerautomorphism as a subgroup of π1(M, γ).

4A categorical way of phrasing this is as follows. Let γ denote the category whose objects are points p ∈ γ,with a single morphism p → p′ for all p, p′ ∈ γ. The fundamental group is a functor π1(M, ·) : γ → Groups,where the morphism p → p′ is sent to the isomorphism π1(M,p) → π1(M,p′) given by the path from p to p′

along γ. Then π1(M,γ) is defined as the limit/colimit of this functor.

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Lemma 2.14. Let M be a directed quasicylinder. Then the pair (S(M),≤) is a partially-ordered set. That is, for all F1,F2,F3 ∈ S(M) we have:

1. (reflexivity) F1 ≤ F1.2. (antisymmetry) F1 ≤ F2 ≤ F1 =⇒ F1 = F2.3. (transitivity) F1 ≤ F2 ≤ F3 =⇒ F1 ≤ F3.

Proof. By Lemma 2.11, it suffices to work in the PL category.Reflexivity is obvious.For transitivity, suppose F1 ≤ F2 ≤ F3. Then pick representatives F1, F2, F

′2, F

′3 such that

F1 ≤ F2 and F ′2 ≤ F ′

3. By Lemma 2.3, there is a PL isotopy from F2 to F ′2. Applying this

isotopy to F1 produces F ′1 ≤ F ′

2 ≤ F ′3.

For antisymmetry, suppose F1 ≤ F2 and F2 ≤ F1. Pick PL representatives F1 ≤ F2 ≤F ′1 ≤ F ′

2 (this is possible by the argument used for transitivity). Now pick a PL arc γ fromone end ofM to the other which intersects each of F1, F

′1, F2, F

′2 exactly once. Since the maps

π1(F1), π1(F′1), π1(F2), π1(F

′2) → π1(M, γ) are injective, we will identify each of the groups

on the left with its image in π1(M, γ). By Lemma 2.12, we have π1(Fi) = π1(F′i ) under this

identification.By van Kampen’s theorem we have:

π1(M, γ) = π1((M \ F2)−, γ) ∗π1(F2)

π1((M \ F2)+, γ) (2.1)

Since F1 ≤ F2 ≤ F ′1, we have π1(F1) ⊆ π1((M \ F2)−, γ) and π1(F

′1) ⊆ π1((M \ F2)+, γ).

Since π1(F1) = π1(F′1), we have π1(F1) ⊆ π1((M \F2)−, γ)∩π1((M \F2)+, γ). It follows from

the properties of the amalgamated product (2.1) that this intersection is just π1(F2). Thuswe have π1(F1) ⊆ π1(F2). A symmetric argument shows the reverse inclusion, so we haveπ1(F1) = π1(F2). Now an easy obstruction theory argument shows F1 and F2 are homotopic(using the fact that π2(M) = 0 by the sphere theorem [29]). Thus F1 = F2.

Lemma 2.15. LetM be a directed quasicylinder. Let F be a bicollared embedded π1-injectivesurface. Then the natural map ψ : S((M \F )−) → S(M) is a bijection of S((M \F )−) withthe set

{

G ∈ S(M)∣

∣ G ≤ [F ]}

. Furthermore, this bijection satisfies G1 ≤ G2 ⇐⇒ ψ(G1) ≤ψ(G2).

Proof. By Lemma 2.11, it suffices to work in the PL category.Certainly the image of ψ is contained in

{

G ∈ S(M)∣

∣ G ≤ [F ]}

. If G ≤ [F ], thenthere are representatives G′ ≤ F ′. By Lemma 2.3, there is a PL isotopy sending F ′ toF . Applying this isotopy to G′ gives a representative G ≤ F . Thus the image of ψ isexactly

{

G ∈ S(M)∣

∣ G ≤ [F ]}

. To prove that ψ is injective, suppose G1, G2 ≤ F aretwo π1-injective embedded surfaces which are homotopic in M . Pick an arc γ from oneend of M to the other which intersects G1, F exactly once. Since G1, G2 are homotopic,Lemma 2.12 gives canonical maps π1(G1), π1(G2) → π1(M, γ) with the same image. Nowthese both factor through π1((M \ F )−, γ) → π1(M, γ), which is injective since π1(F ) →π1((M \F )±, γ) are injective. Thus π1(G1), π1(G2) → π1((M \F )−, γ) have the same image,so the same obstruction theory argument used in the proof of Lemma 2.14 implies thatG1, G2 are homotopic in (M \ F )−. Thus ψ is injective.

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Now it remains to show that ψ preserves ≤. The only nontrivial direction is to showthat ψ(G1) ≤ ψ(G2) =⇒ G1 ≤ G2. If ψ(G1) ≤ ψ(G2), then we have representativesG′

1 ≤ G′2 ≤ F ′ (by the argument for transitivity in Lemma 2.14). Now by Lemma 2.3 there

is a PL isotopy from F ′ to F , and applying this isotopy to G′1, G

′2, we get G1 ≤ G2 ≤ F .

Since ψ is injective, we have [Gi] = Gi in S((M \ F )−). Thus G1 ≤ G2.

Lemma 2.16. Let M be a directed quasicylinder. Let F1 ≤ F2 be two bicollared embeddedπ1-injective surfaces. Then the natural map ψ : S((M \ F1)+ ∩ (M \ F2)−) → S(M) is abijection of S((M \ F1)+ ∩ (M \ F2)−) with

{

G ∈ S(M)∣

∣ [F1] ≤ G ≤ [F2]}

. Furthermore,this bijection satisfies G1 ≤ G2 ⇐⇒ ψ(G1) ≤ ψ(G2).

Proof. This is just two applications of Lemma 2.15.

Lemma 2.17. Let M be a PL directed quasicylinder. Let G ⊆ M be any PL embeddedsurface (not necessarily π1-injective) separating the two ends of M . Then there exists a classG ∈ S(M) such that for all F ∈ S(M) with a representative F ≤ G (resp. F ≥ G), we haveF ≤ G (resp. F ≥ G).

Proof. As long as G is compressible, we can perform the following operation. Do some com-pression on G (this may disconnect G), and pick one of the resulting connected componentswhich is nonzero in H2(M) to keep. This operation does not destroy the property that Gcan be isotoped to lie in (M \ F )+ (resp. (M \ F )−) for an incompressible surface F . Sinceeach step decreases the genus of G, we eventually reach an incompressible surface, which bythe loop theorem is π1-injective, and thus defines a class G ∈ S(M).

Lemma 2.18. Let M be a directed quasicylinder. Let A ⊆ S(M) be a finite set. Then thereexist elements A−,A+ ∈ S(M) such that A− ≤ A ≤ A+ for all A ∈ A.

Proof. By Lemma 2.11, it suffices to work in the PL category.Pick PL transverse representatives A1, . . . , An of all the surfaces in A. Let ((M \A1)± ∩

· · · ∩ (M \An)±)0 denote the unique unbounded component of (M \A1)± ∩ · · · ∩ (M \An)±,and define A± = ∂(((M \A1)± ∩ · · · ∩ (M \An)±)0). It is easy to see that A± are connectedand separate the two ends of M . Now apply Lemma 2.17 to A− and A+ to get the desiredclasses A−,A+ ∈ S(M).

Lemma 2.19 (suggested by Ian Agol [1]). Let M be a directed quasicylinder. Then thepartially-ordered set (S(M),≤) is a lattice. That is, for F1,F2 ∈ S(M), the following set hasa least element:

X(F1,F2) ={

H ∈ S(M)∣

∣ F1,F2 ≤ H}

(2.2)

(and the same holds in the reverse ordering).

Proof. By Lemma 2.11, it suffices to work in the PL category.First, let us deal with the case where M has tame ends, that is M is the interior of a

compact PL manifold with boundary (M, ∂M). Equip (M, ∂M) with a smooth structureand a smooth Riemannian metric so that the boundary is convex.

Now let us recall some facts about area minimizing surfaces. The standard existence the-ory of Schoen–Yau [35] and Sacks–Uhlenbeck [33, 34] gives the existence of area minimizing

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maps in any homotopy class of π1-injective surfaces. By Osserman [28, 27] and Gulliver [11],such area minimizing maps are immersions. Now Freedman–Hass–Scott [7] have proved the-orems to the effect that area minimizing representatives of π1-injective surfaces in irreduciblethree-manifolds have as few intersections as possible given their homotopy class. Thoughthey state their main results only for closed irreducible three-manifolds, they remark [7,p634 §7] that their results remain true for compact irreducible three-manifolds with convexboundary.

Suppose F1,F2 ∈ S(M); let us show that X(F1,F2) has a least element. Pick areaminimizing representatives F1, F2 in the homotopy classes of F1,F2. By [7, p626 Theorem5.1], F1, F2 are embedded. Let ((M \ F1)+ ∩ (M \ F2)+)0 denote the unique unboundedcomponent of (M \ F1)+ ∩ (M \ F2)+, and define F = ∂(((M \ F1)+ ∩ (M \ F2)+)0). It iseasy to see that F is connected and separates the two ends of M . Note that F1, F2 can beisotoped to lie in (M \ F )−. Furthermore, if G ∈ X(F1,F2) (and G 6= F1,F2), then G hasa least area representative G, which by [7, p630 Theorem 6.2] is disjoint from F1 and F2.Thus F1, F2 ≤ G, so F ≤ G. Thus we may apply Lemma 2.17 to F and produce an elementF ∈ S(M) which is a least element of X(F1,F2). This finishes the proof for M with tameends.

Now let us treat the case of general M . Consider the following sets:

X(F1,F2;G) ={

H ∈ S(M)∣

∣ F1,F2 ≤ H ≤ G}

⊆ X(F1,F2) (2.3)

We claim that it suffices to show that if G ∈ X(F1,F2), thenX(F1,F2;G) has a least element.To see this, we argue as follows. If G1,G2 ∈ X(F1,F2) and G1 ≤ G2, then it is easy to seethat the natural inclusion X(F1,F2;G1) → X(F1,F2;G2) sends the least element of thedomain to the least element of the target (assuming both sets have least elements). Sincefor every G1,G2, there exists G3 greater than both (by Lemma 2.18), we see that the leastelements of all the sets {X(F1,F2;G)}G∈X(F1,F2) are actually the same element F ∈ X(F1,F2).We claim that this F is a least element of X(F1,F2). This is trivial: if G ∈ X(F1,F2), thenF is a least element of X(F1,F2;G), so a fortiori F ≤ G. Thus it suffices to show that eachset X(F1,F2;G) has a least element.

Let us show that X(F1,F2;G) has a least element. By Lemma 2.18 there exists F0 ∈S(M) so that F0 ≤ F1,F2. Now pick PL representatives F0, G of F0,G with F0 ≤ G. LetM0 = (M \F0)+ ∩ (M \G)−, which is a directed quasicylinder with tame ends. Thus by thecase dealt with earlier, S(M0) is a lattice, so F1,F2 have a least upper bound in S(M0). ByLemma 2.16, S(M0) → S(M) is an isomorphism onto the subset

{

F ∈ S(M)∣

∣ F0 ≤ F ≤ G}

.Thus we get the desired least element of X(F1,F2;G).

Remark 2.20. It should be possible to prove an existence result for area minimizing surfacesin any DIFF quasicylinder (with appropriate conditions on the Riemannian metric near theends). In that case, the argument used for tame quasicylinders would apply in general (theresults of Freedman–Hass–Scott [7] extend as long as one has the existence of area minimizingrepresentatives of π1-injective surfaces in M).

Remark 2.21. Jaco–Rubinstein [13] have developed a theory of normal surfaces in triangu-lated three-manifolds analogous to that of minimal surfaces in smooth three-manifolds with

9

a Riemannian metric. In particular, they have results analogous to those of Freedman–Hass–Scott [7]. Thus it is likely that we could eliminate entirely the use of the smooth categoryand the results in minimal surface theory for the proof of Lemma 2.19.

3 Tools applicable to arbitrary open/closed subsets of

manifolds

In our study of a hypothetical action of Zp by homeomorphisms on a three-manifold, it willbe essential to study certain properties of orbit sets (for example, their homology). However,we do not have the luxury of assuming such sets are at all well behaved; the most we canhope for is that we will be able to construct Zp-invariant sets which are either open or closed.Nevertheless, Cech cohomology is still a reasonable object in such situations. The purpose ofthis section is to develop an elementary theory centered on Alexander duality for arbitraryopen and closed subsets of a manifold. We use these tools in the proof of Theorem 1.5,specifically in Sections 4.3–4.4 to construct the set Z and to study its properties.

In this (and all other) sections, we always take homology and cohomology with integercoefficients. We denote by H∗ and H∗ singular homology and cohomology, and we let H∗

denote Cech cohomology.

3.1 Cech cohomology

Lemma 3.1. For a compact subset X of a manifold, the natural map below is an isomor-phism:

limU⊇XU open

H∗(U)∼−→ H∗(X) (3.1)

Proof. This is due to Steenrod [39]; see also Spanier [38, p419], the key fact being that Cechcohomology satisfies the “continuity axiom”.

Lemma 3.2. If X and Y are two compact subsets of a manifold, then there is a (Mayer–Vietoris) long exact sequence:

· · · → H∗(X ∪ Y ) → H∗(X)⊕ H∗(Y ) → H∗(X ∩ Y ) → H∗+1(X ∪ Y ) → · · · (3.2)

Proof. For arbitrary open neighborhoods U ⊇ X and V ⊇ Y , we have a Mayer–Vietorissequence of singular cohomology for U and V . Taking the direct limit over all U and V givesa sequence of the form (3.2) by Lemma 3.1, and it is exact since the direct limit is an exactfunctor.

3.2 Alexander duality

Alexander duality for subsets of Sn is most naturally stated using reduced homology andcohomology, which we denote by H .

10

Lemma 3.3. Let X ⊆ Sn be a compact set. Then:

˜H∗(X) = Hn−1−∗(Sn \X) (3.3)

Proof. If U ⊆ Sn is an open subset with smooth boundary, then it is well-known that thereis a natural isomorphism H∗(U)

∼−→ Hn−1−∗(Sn \ U). Furthermore, for U ⊇ V the following

diagram commutes:H∗(U)

∼−−−→ Hn−1−∗(Sn \ U)

y

y

H∗(V )∼

−−−→ Hn−1−∗(Sn \ V )

(3.4)

Now consider the direct limit over all such open sets U ⊇ X to get an isomorphism of thedirect limits. This family of sets U forms a final system of neighborhoods of X , so by Lemma

3.1 the direct limit on the left is ˜H∗(X). The direct limit on the right is Hn−1−∗(Sn \X) byelementary properties of singular homology.

3.3 The plus operation

If we have a bounded subset X ⊆ R3, we want to be able to consider “X union all of thebounded connected components of R3 \X” (we think of this operation as a way to simplifythe set X , which could be very wild). The following definition makes this precise.

Definition 3.4. For a bounded set X ⊆ R3, we define:

X+ =⋂

open U⊇XU boundedH2(U)=0

U (3.5)

By Lemma 3.3, for bounded open sets U ⊆ X , we have H2(U) = 0 ⇐⇒ H0(R3 \ U) = Z.With this reformulation, it is easy to see that if U and V both appear in the intersection,then so does U ∩ V . Let us call X 7→ X+ the plus operation.

Lemma 3.5. The plus operation satisfies the following properties:i. X ⊆ Y =⇒ X+ ⊆ Y +.ii. X++ = X+.iii. If X is closed and bounded, then R3 \X+ is the unbounded component of R3 \X.iv. If X is closed and bounded, then X = X+ ⇐⇒ H2(X) = 0 ⇐⇒ R3\X is connected.

Proof. Both (i) and (ii) are immediate from the definition.For (iii), argue as follows. Let V denote the unbounded component of R3 \X . As V is

open and connected, it has an exhaustion by closed connected subsets V =⋃∞

i=1 Vi, whereV1 ⊆ V2 ⊆ · · · and R3 \ Vi is bounded for all i. By Lemma 3.3, H2(R3 \ Vi) = 0, and thusX+ ⊆

⋂∞i=1(R

3 \ Vi) = R3 \ V . It remains to show the reverse inclusion, namely that if Wis a bounded component of R3 \X , then W ⊆ X+. Suppose U ⊇ X is open, bounded, andH2(U) = 0. Then R3 \U = (R3 \ (U ∪W ))∪ (W \U), which is a disjoint union of closed sets.

11

However R3 \ U is connected and R3 \ (U ∪W ) is unbounded so a fortiori it is nonempty.Thus we must have W \ U = ∅, that is U ⊇W .

For (iv), argue as follows. Lemma 3.3 gives H2(X) = 0 ⇐⇒ R3 \X is connected. By(iii), we have X = X+ ⇐⇒ R3 \X is connected.

Lemma 3.6. Suppose X ⊆ R3 is closed and bounded with X = X+. If {Uα}α∈A is a finalcollection of open sets containing X, then so is {U+

α }α∈A.

Proof. Since {Uα}α∈A is final, to show that {U+α }α∈A is final it suffices to show that for every

α ∈ A, there exists β ∈ A such that U+β ⊆ Uα.

Thus let us suppose α ∈ A is given. As in the proof of Lemma 3.5(iii), there exists anexhaustion by closed connected subsets R3 \X =

⋃∞i=1 Vi, where V1 ⊆ V2 ⊆ · · · and R3 \ Vi

is bounded for all i. Then for sufficiently large i, we have R3 \ Vi ⊆ Uα. On the other hand,for every i there exists β such that Uβ ⊆ R3 \ Vi. But now U+

β ⊆ (R3 \ Vi)+ = R3 \ Vi ⊆ Uα

as needed.

Definition 3.7. Let X be a topological space. We say X has dimension zero iff wheneverx ∈ U ⊆ X with U open, there exists a clopen (closed and open) set V ⊆ X with x ∈ V ⊆ U .It is immediate that any subspace of a space of dimension zero also has dimension zero.

Remark 3.8. If M is a manifold with a continuous action of Zp, then every orbit is eithera finite discrete set (if the stabilizer is pkZp) or a cantor set (if the stabilizer is trivial). Inparticular, every orbit has dimension zero. It follows that if A ⊆ M is any finite set, thenZpA has dimension zero, as does any subset thereof.

Lemma 3.9. IfM is a manifold of dimension n ≥ 2 and X ⊆M is closed and has dimensionzero, then the map H0(M \X) → H0(M) is an isomorphism.

Proof. First, let us deal with the case M = Sn (thus X is compact). Pick a metric on Sn andfix ǫ > 0. Since X has dimension zero and is compact, we can cover X with finitely manyclopen sets Ui ⊆ X (1 ≤ i ≤ N) of diameter ≤ ǫ. Now {Ui \

⋃

1≤j<i Uj}Ni=1 is a cover of X by

disjoint open sets of diameter ≤ ǫ. Since ǫ > 0 was arbitrary, this implies that H i(X) = 0

for i > 0. Thus by Lemma 3.3, H0(Sn \X) = ˜Hn−1(X) = 0 (since n ≥ 2), which is sufficient.Second, let us deal with the case M = Rn. Define X = X ∪ {∞} ⊆ Rn ∪ {∞} = Sn. We

claim that X has dimension zero; to see this, it suffices to produce small clopen neighborhoodsof ∞ ∈ X . Fix R <∞. Since X has dimension zero and X ∩B(0, R) is compact, there existfinitely many clopen sets Ui ⊆ X (1 ≤ i ≤ N) of diameter ≤ 1 with X ∩B(0, R) ⊆

⋃Ni=1Ui.

Then X \⋃N

i=1 Ui is clopen in X and contains ∞. Since R < ∞ was arbitrary, we getarbitrarily small clopen neighborhoods of ∞ ∈ X . Hence X has dimension zero, so by thecase M = Sn dealt with above, we have H0(Rn \X) = H0(Sn \ X) = Z, which is sufficient.

Now let us deal with the case of generalM . It suffices to show that H0(M) → H0(M \X)is an isomorphism (recall that H0 is just the group of locally constant maps to Z). Nowconsider any p ∈ M , and pick an open neighborhood p ∈ U ∼= Rn. Since X has dimensionzero, so does U ∩ X . Thus by the case M = Rn dealt with above, U \X is connected andnonempty. Thus any locally constant function M \ X → Z can be extended uniquely to alocally constant function M → Z, as needed.

12

4 Nonexistence of faithful actions of Zp on three-manifolds

Proof of Theorem 1.5. Let M be a connected three-manifold and suppose there is a contin-uous injection of topological groups Zp → Homeo(M).

4.1 Step 1: Reduction to a local problem in R3

Let B(r) denote the open ball of radius r centered at the origin 0 in R3. Fix a small positivenumber η = 2−10. In this section, we reduce to the case where M is an open subset of R3

such that:i. B(1) ⊆ M ⊆ B(1 + η).ii. dR3(x, αx) ≤ η for all x ∈M and α ∈ Zp.iii. No subgroup pkZp fixes an open neighborhood of 0.

This reduction is valid in any dimension.

Definition 4.1. We say that an action of a group G on a topological space X is locally offinite order at a point x ∈ X if and only if some subgroup of finite index G′ ≤ G fixes anopen neighborhood of x. The action is (globally) of finite order if and only if some subgroupof finite index G′ ≤ G fixes all of X .

Newman’s theorem [24, p6 Theorem 2] (see also Dress [6, p204 Theorem 1] or Smith [37])implies that on a connected manifold, a group action which is (everywhere) locally of finiteorder is globally of finite order. Thus if Zp → Homeo(M) is injective, then there exists apoint m ∈ M where the action is not locally of finite order. In other words, no subgrouppkZp fixes an open neighborhood of m.

Now fix some homeomorphism between B(3) ⊆ R3 and an open set in M containing m,such that 0 is identified with m. Note that (by continuity of the map Zp → Homeo(M))for sufficiently large k, we have pkZpB(2) ⊆ B(3) and dR3(x, αx) ≤ η for all x ∈ B(2) andα ∈ pkZp. Thus the action of pkZp (which is isomorphic to Zp) on M

′ := pkZpB(1) satisfies(i), (ii), and (iii) above. Thus it suffices to replace (M,Zp) with (M ′, pkZp) and derive acontradiction.

Henceforth we assume that M is an open subset of R3 satisfying (i), (ii), and (iii) above.

4.2 Step 2: An invariant metric and the plus operation in M

Equip M with the following Zp-invariant metric, which induces the same topology as dR3 :

dinv(x, y) :=

∫

Zp

dR3(αx, αy) dµHarr(α) (4.1)

We will never mention dinv again explicitly, but we will often write N invǫ for the open ǫ-

neighborhood in M with respect to dinv. Note that if X ⊆ M is Zp-invariant, then so isN inv

ǫ (X).

Lemma 4.2. If X ⊆ B(1) ⊆ M , then X+ ⊆ B(1) ⊆ M . If in addition X is Zp-invariant,then so is X+.

13

Figure 1: Diagram of K0 inside B(1) ⊆M .

Proof. The first statement is clear since B(1) appears in the intersection (3.5) defining X+.Now suppose that in addition X is Zp-invariant. As remarked in Definition 3.4, the

collection of open sets appearing in the intersection (3.5) is closed under finite intersection.In particular, we could consider just those U contained in B(1). Thus a fortiori X+ is theintersection of all open sets U with X ⊆ U ⊆ M and H2(U) = 0. The set of such U ispermuted by Zp, so X

+ is Zp-invariant.

4.3 Step 3: Construction of an interesting compact set Z ⊆M

In this section, we construct a compact Zp-invariant set Z ⊆ M such that:1. On a coarse scale, Z looks like a handlebody of genus two.2. The action of Zp on H1(Z) is nontrivial.

We follow a natural strategy to construct the set Z. We first let K be the orbit under Zp

of a closed handlebody of genus two, and we then construct a set L as the orbit under Zp

of a small arc connecting two points on the boundary of K. Then we let Z = K ∪ L. Thisstrategy is complicated by the fact that the orbit set K is a priori very wild, and we willneed to investigate the connectedness properties of it and other sets in order to make theconstruction work. We will take advantage of the tools we have developed in Section 3.3 andthe setup from Sections 4.1–4.2. Even with this preparation, though, careful verification ofeach step takes us a while.

Let x0 ∈ B(η) \ FixZp (such an x0 exists by Step 1(iii)).

Definition 4.3 (see Figure 1). Let K0 ⊆ B(1) be a closed handlebody of genus two whoseunique point of lowest z-coordinate is x0. Note that the closed η-neighborhood (in theEuclidean metric) of K0 is a larger handlebody isotopic to K0. Let K = (ZpK0)

+.

We think of K as being illustrated essentially by Figure 1; it’s just K0 plus some wildfuzz of width η. Note that K is compact, Zp-invariant (by Lemma 4.2), and K = K+. Also,K is connected; actually, we will need the following stronger fact.

Lemma 4.4. If A ⊆ ∂K is any finite set, then K \ ZpA is path-connected.

14

Proof. By definition, K◦0 is path-connected. Now the action of Zp is within η = 2−10 of the

identity, so Zp(K◦0 ) is also path-connected.

Now suppose x ∈ ZpK0 \ ZpA. Then there exists α ∈ Zp so that α−1x ∈ K0. Fromα−1x there is clearly a path inwards to K◦

0 , and this is disjoint from ZpA since ZpA ⊆ ∂K.Translating this path by α shows that ZpK0 \ ZpA is path-connected.

Now suppose x ∈ K \ ZpA. If x /∈ ZpK0, then let V denote the (open) connectedcomponent of K \ ZpK0 containing x. By Remark 3.8 and Lemma 3.9, we can find a pathfrom x to infinity in R3 \ ZpA. There is a time when this path first hits ∂V ⊆ ZpK0, thusgiving us a path from x to ZpK0 contained in K \ZpA. Thus K \ZpA is path-connected, aswas to be shown.

Lemma 4.5. There exist points x1, x2 ∈ K with x1 /∈ FixZp and Zpx1∩Zpx2 = ∅, along witha compact Zp-invariant set L ⊆M of diameter ≤ 4η so that L = L+, K ∩L = Zpx1 ∪Zpx2,and L contains a path from x1 to x2.

Proof. Let x1 ∈ K be any point of lowest z-coordinate in K. We claim that x1 /∈ FixZp.First, observe that since x1 ∈ K = (ZpK0)

+ is a point of lowest z-coordinate, necessarilyx1 ∈ ZpK0. If x1 ∈ K0, then x1 = x0 (recall from Definition 4.3 that x0 is the uniquepoint of lowest z-coordinate in K0), and by definition x0 /∈ FixZp. On the other hand, ifx1 ∈ (ZpK0) \K0, then certainly x1 /∈ FixZp. Thus x1 /∈ FixZp.

Since x1 ∈ ZpK0, we may pick x′2 ∈ Zp(K◦0 ) ⊆ K◦ which is arbitrarily close to x1.

Specifically, let us fix x′2 ∈ K◦∩B(x1, η). Note that x′2 /∈ Zpx1 since x1 /∈ K◦. Now we claim

that there exists a continuous path γ : [0, 1] → B(x1, η) such that:i. γ(0) = x1.ii. γ(1) = x′2.iii. γ−1(Zpx1) = {0}.iv. 0 is not a limit point of γ−1(K).

To construct such a path γ, argue as follows. First, define γ : [0, 12] → B(x1, η) to be some

path straight downward from x1 (this takes care of (i)). Since x1 is a point of smallestz-coordinate in K ⊇ Zpx1, this also takes care of (iv) and is consistent with (iii). Now byRemark 3.8 and Lemma 3.9, we know B(x1, η) \ Zpx1 is path-connected, so we can find apath γ : [1

2, 1] → B(x1, η) \ Zpx1 from γ(1

2) to x′2 (this takes care of (ii) and (iii)). Splicing

these two functions together gives γ : [0, 1] → B(x1, η) satisfying (i), (ii), (iii), and (iv).Now by (ii) and (iv) we have that t := min(γ−1(K) \ {0}) exists and is positive. Let

x2 = γ(t), and define L0 = γ([0, t]). By (iii), we have Zpx1∩Zpx2 = ∅. Define L = (ZpL0)+,

which is Zp-invariant by Lemma 4.2. By construction, L contains a path from x1 to x2.Certainly L0 ⊆ B(x1, η) so L ⊆ B(x1, 2η) by Step 1(ii), and so L has diameter ≤ 4η. Itremains only to show that K ∩ L = Zpx1 ∪ Zpx2 (certainly the containment ⊇ is given bydefinition).

Note that K \ ZpL0 = K \ (Zpx1 ∪ Zpx2), which by Lemma 4.4 is path-connected. ThusK \ZpL0 lies in a single connected component of R3 \ZpL0. Now ZpL0 has diameter ≤ 4η asremarked above, and K \ZpL0 contains a large handlebody, so it must lie in the unboundedcomponent of R3 \ ZpL0. Hence K \ (Zpx1 ∪ Zpx2) is disjoint from (ZpL0)

+ = L.

Definition 4.6. Fix x1, x2, and L as exist by Lemma 4.5, and define Z = K ∪ L.

15

4.4 Step 4: The cohomology of Z

Lemma 4.7. Z = Z+, and the action of Zp on H1(Z) is nontrivial.

Proof. The key to proving both statements is the following Mayer–Vietoris sequence (whichexists and is exact by Lemma 3.2):

· · · → H∗(Z) → H∗(K)⊕ H∗(L) → H∗(K ∩ L) → H∗+1(Z) → · · · (4.2)

Note that Zp acts on all of these groups and that the maps are Zp-equivariant.By Lemma 4.5, K ∩L = Zpx1 ∪Zpx2, so by Remark 3.8, H i(K ∩L) = 0 for i > 0. Thus

we have H2(Z) = H2(K)⊕ H2(L). Now applying Lemma 3.5(iv) twice shows that K = K+

and L = L+ together imply that Z = Z+.Now let us show that Zp acts on H1(Z) nontrivially. We use the exact sequence:

H0(K)⊕ H0(L) → H0(K ∩ L) → H1(Z) (4.3)

Note that H0 is just the group of locally constant functions to Z. Thus it suffices to exhibita locally constant function q : K ∩ L→ Z and an element α ∈ Zp such that αq − q is not inthe image of H0(K)⊕ H0(L). Let us define:

q(r) =

{

1 r ∈ pZpx1

0 r ∈⋃

a∈(Z/p)\{0}(a+ pZp)x1 ∪ Zpx2(4.4)

(observe by Lemma 4.5 that x1 /∈ FixZp, so the sets on the right hand side are indeeddisjoint). Let α ∈ Zp be congruent to 1 mod p. Then we have:

(αq − q)(r) =

−1 r ∈ pZpx1

1 r ∈ (1 + pZp)x1

0 r ∈⋃

a∈(Z/p)\{0,1}(a+ pZp)x1 ∪ Zpx2

(4.5)

Now by Lemma 4.5, x1, x2 are in the same component of L, and by Lemma 4.4 (takingA = ∅), they are in the same component of K. Thus every function in the image ofH0(L)⊕ H0(K) assigns the same value to x1 and x2. Clearly this is not the case for αq− q,so we are done.

We just proved that Z = Z+, so by Lemma 3.6, N invǫ (Z)+ is a final system of neighbor-

hoods of Z. Thus by Lemma 3.1 we have H1(Z) = lim−→

H1(N invǫ (Z)+) as abelian groups with

an action of Zp (Zp acts on the latter since N invǫ (Z)+ is Zp-invariant by Lemma 4.2). Since

the action on the limit group is nontrivial, the following definition makes sense.

Definition 4.8. Fix ǫ ∈ (0, η) such that the Zp action on the image of the mapH1(N invǫ (Z)+) →

H1(Z) is nontrivial. Denote by (N invǫ (Z)+)0 the connected component containing Z, and

define U = (N invǫ (Z)+)0 \ Z, which is Zp-invariant by Lemma 4.2.

16

4.5 Step 5: Special elements of S(U)

Lemma 4.9. The set U is a quasicylinder in the sense of Definition 2.4.

Proof. We have R3 \ U = Z ∪ (R3 \ (N invǫ (Z)+)0); the latter two sets are connected and

disjoint, so by Lemma 3.3, we have H2(U) = Z.Suppose F ⊆ U is a PL embedded sphere; then int(F ) (the bounded component of R3\F )

is a ball [2]. If Z ⊆ int(F ), then we have a factorization H1(N invǫ (Z)+) → H1(int(F )) →

H1(Z), which implies the composition is trivial, contradicting the definition of U . ThusZ * int(F ), so int(F ) ⊆ U , that is F bounds ball in U . Thus U is irreducible.

Certainly U has at least two ends, and since H2(U) = Z it has at most two ends.Since U is an open subset of R3, it is orientable.

Recall the definition of (S(U),≤) from Section 2. Certainly the action of Zp on U inducesan action on STOP(U).

Lemma 4.10. There exists an element F ∈ S(U) which is fixed by Zp.

Proof. Observe that if F is any surface in U , then for sufficiently large k, we have that αFis homotopic to F for all α ∈ pkZp. Thus Zp acts on S(U) with finite orbits. Now any groupacting with finite orbits on a nonempty lattice has a fixed point, namely the least upperbound of any orbit. Recall that S(U) is nonempty by Remark 2.8.

Definition 4.11. Fix a PL π1-injective surface F ⊆ U such that [F ] ∈ S(U) is fixed by Zp.Denote by int(F ) and ext(F ) the two connected components of R3 \ F .

Lemma 4.12. The Zp action on U induces a homomorphism Zp → MCG(F ), as wellas actions on all the homology and cohomology groups appearing in (4.6)–(4.9). Theseactions are compatible with the maps in (4.6)–(4.9), as well as with the map MCG(F ) →Aut(H1(F )).

H1(N invǫ (Z)+) −−−→ H1(int(F )) −−−→ H1(Z)

y

H1(F )

(4.6)

H1(Z) → H1(int(F )) (4.7)

H1(M \N invǫ (Z)+) → H1(ext(F )) (4.8)

H1(F )∼−→ H1(int(F ))⊕H1(ext(F )) (4.9)

Proof. For every α ∈ Zp, we know by Lemma 2.3 that there is a compactly supportedisotopy of U sending αF to F . Now such an isotopy can clearly be extended to all of M asthe constant isotopy on M \ U . Thus for every α ∈ Zp, let us pick an isotopy:

ψtα :M →M (t ∈ [0, 1]) (4.10)

so that ψ0α = idM , ψ1

α(αF ) = F , and ψtα(x) = x for x ∈M \ U . Denote by Tα :M → M the

action of α ∈ Zp.

17

Now observe that for every α ∈ Zp, the map ψ1α ◦ Tα fixes Z, N inv

ǫ (Z)+, and F . Thusthe map ψ1

α ◦ Tα induces an automorphism of each of the diagrams (4.6)–(4.9) and gives acompatible element of MCG(F ). It remains only to show that this is a homomorphism fromZp.

We need to show that for all α, β ∈ Zp, the two maps ψ1β ◦ Tβ ◦ ψ1

α ◦ Tα and ψ1βα ◦ Tβα

induce the same action on (4.6)–(4.9) and give the same element of MCG(F ). It of coursesuffices to show that ψ1

β ◦Tβ ◦ψ1α ◦Tα ◦ (ψ

1βα ◦Tβα)

−1 induces the trivial action on (4.6)–(4.9)and gives the trivial element of MCG(F ). Now we write:

ψ1β ◦ Tβ ◦ ψ

1α ◦ Tα ◦ (ψ1

βα ◦ Tβα)−1 = ψ1

β ◦ Tβ ◦ ψ1α ◦ Tα ◦ T−1

βα ◦ (ψ1βα)

−1

= ψ1β ◦ Tβ ◦ ψ

1α ◦ T−1

β ◦ (ψ1βα)

−1 (4.11)

This is the identity map on Z and on M \ N invǫ (Z)+, so the action on their (co)homology

is trivial. The map (4.11) is isotopic to the identity map via ψtβ ◦ Tβ ◦ ψt

α ◦ T−1β ◦ (ψt

βα)−1

for t ∈ [0, 1], which only moves points in a compact subset of U . Thus the action on thecohomology of N inv

ǫ (Z)+ is trivial as well. By Lemma 2.13, this isotopy induces the trivialelement in MCG(F ), and this means that the action on the (co)homology of F , int(F ), andext(F ) is trivial as well.

Lemma 4.13. The map Zp → MCG(F ) annihilates an open subgroup of Zp and has non-trivial image.

Proof. The action of Zp on U is continuous, so for sufficiently large k, we have that αF andF are homotopic as maps F → U for all α ∈ pkZp. Thus the homomorphism Zp → MCG(F )annihilates a neighborhood of the identity in Zp.

To prove that the image is nontrivial, it suffices to show that the Zp action on H1(F ) isnontrivial. For this, consider equation (4.6). By Definition 4.8, there exists an element ofH1(N inv

ǫ (Z)+) whose image in H1(Z) is not fixed by Zp. Thus the action of Zp on H1(int(F ))

is nontrivial. Now the vertical map H1(int(F )) → H1(F ) is injective (otherwise the Mayer–Vietoris sequence applied to R3 = int(F ) ∪F ext(F ) would imply H1(R3) 6= 0). Thus theaction of Zp on H1(F ) is nontrivial as well.

Lemma 4.14. There is a rank four submodule of H1(F )Zp on which the intersection form

is:(

0 1−1 0

)

⊕

(

0 1−1 0

)

(4.12)

Proof. There are two obvious loops in K0 (Definition 4.3) generating its homology, and twoobvious dual loops in M \N inv

ǫ (Z)+. Since the Zp action is within η = 2−10 of the identity, itis easy to see that their classes in homology are fixed by Zp. Thus we have well-defined classesα1, α2 ∈ H1(Z)

Zp and β1, β2 ∈ H1(M \ N invǫ (Z)+)Zp with linking numbers lk(αi, βj) = δij .

Now the maps from equations (4.7), (4.8), and the isomorphism (4.9) give us correspondingelements α1, α2, β1, β2 ∈ H1(F )

Zp. The intersection form on H1(F ) coincides with the linkingpairing between H1(int(F )) and H1(ext(F )). Thus the intersection form on the submoduleof H1(F )

Zp generated by α1, α2, β1, β2 is indeed given by (4.12).

18

By Lemma 4.13, the image of Zp in MCG(F ) is a nontrivial cyclic p-group. It has a(unique) subgroup isomorphic to Z/p, and by Lemma 4.14 this subgroup Z/p ⊆ MCG(F )has the property that H1(F )

Z/p has a submodule on which the intersection form is given by(4.12). This contradicts Lemma 5.1 below, and thus completes the proof of Theorem 1.5.

Remark 4.15. One expects that if we make U thick enough around most of Z (but still thinnear L), then away from L the surface F can be made to look like a punctured surface ofgenus two. However, it is not clear how to prove this stronger, more geometric propertyabout F .

If we could prove this, then the final analysis is more robust. Instead of homology classesα1, α2, β1, β2 ∈ H1(F )

Zp, we now have simple closed curves α1, α2, β1, β2 in F which arefixed up to isotopy by the Zp action. Now the finite image of Zp in MCG(F ) is realizedas a subgroup of Isom(F, g) for some hyperbolic metric g on F . A hyperbolic isometryfixing α1, α2, β1, β2 fixes their unique length-minimizing geodesic representatives, and thusfixes their intersections, forcing the isometry to be the identity map. This contradicts thenontriviality of the homomorphism Zp → MCG(F ). As the reader will readily observe, itis enough that α1, β1 be fixed, so we could let K0 be a handlebody of genus one and thisargument would go through.

5 Actions of Z/p on closed surfaces

Lemma 5.1. Let F be a closed connected oriented surface, and let Z/p ⊆ MCG(F ) be somecyclic subgroup of prime order. Then the intersection form of F restricted to H1(F )

Z/p andtaken modulo p has rank at most two.

The proof below follows Symonds [40, pp389–390 Theorem B] and Chen–Glover–Jensen[5, Theorem 1.1], who make explicit a classification due to Nielsen [25] (in fact, Symonds’result actually implies ours for p > 2).

Proof. We seek simply to classify all subgroups Z/p ⊆ MCG(F ) and prove that in each case,the conclusion of the lemma is satisfied. This classification is essentially due to Nielsen [25],who showed that finite cyclic subgroups G ⊆ MCG(F ) are classifed up to conjugacy by their“fixed point data”.

By work of Nielsen [26], any Z/p ⊆ MCG(F ) is realized by a genuine action of Z/p on Fby isometries in some metric (see Thurston [42] or Kerckhoff [15] for a modern perspectiveon this fact and its generalization to any finite group). Now let us switch notation and writeS = F , S = F/(Z/p) (the quotient orbifold), and S for the coarse space of S. The coverS → S is classified by an element α ∈ H1(S,Z/p), which is nonzero since S is connected.Let g denote the genus of S, and n the number of orbifold points of S.

Now for (S,S, α, g, n) as above, we prove the following more precise statement (whichcertainly implies the lemma):

the intersection form on H1(S)Z/p ∼=

{

(

0 p−p 0

)⊕(g−1)⊕ ( 0 1

−1 0 ) n = 0(

0 p−p 0

)⊕gn > 0

(5.1)

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First, let us treat the case n = 0 (so S = S). It is well-known that both compositionsMCG(Σg) → Sp(2g,Z) → Sp(2g,Z/p) are surjective, and that Sp(2g,Z/p) acts transitivelyon the nonzero elements of (Z/p)⊕2g. Thus without loss of generality, it suffices to considerone particular nonzero α ∈ H1(S,Z/p). Thus, let us suppose that α is the Poincare dual of anonseparating simple closed curve ℓ on S. Then the cover S is obtained by cutting S along ℓand gluing together in a circle p copies of the resulting cut surface. Now one easily observesthat there is a direct sum decomposition respecting intersection forms H1(S) ∼= H1(Σ1) ⊕H1(Σg−1)

⊕p, where Z/p acts trivially on H1(Σ1) and acts by rotation of the summandson H1(Σg−1)

⊕p. Then H1(S)Z/p is H1(Σ1) with its usual intersection form, plus a copy of

H1(Σg−1) with its intersection form multiplied by p. Hence equation (5.1) holds in the casen = 0.

Now, let us treat the case n > 0. Label the orbifold points p1, . . . , pn ∈ S, and defineri : H

1(S,Z/p) → Z/p to be the evaluation on a small loop around pi. By construction, theorbifold points are precisely the ramification points of S → S, and hence ri(α) 6= 0 for all i.On the other hand, if D ⊆ S is a disk containing all n orbifold points, then ∂D = ∂(S \ D)is null-homologous in S, so

∑ni=1 ri(α) = 0. Thus in particular n ≥ 2.

Next, let us show how to reduce to the case where either g = 0 or n = 2. So, supposeg ≥ 1 and n ≥ 3, and let D ⊆ S be a disk containing p1, . . . , pn−1. Take D and S \ D andglue in a disk with a single Z/p orbifold point to each, and call the resulting closed orbifoldsS ′ and S ′′ respectively. The class α ∈ H1(S,Z/p) naturally induces α′ ∈ H1(S ′,Z/p) andα′′ ∈ H1(S ′′,Z/p), and so we get covers S ′ and S ′′. Since 〈α, ∂D〉 =

∑n−1i=1 ri(α) = −rn(α) 6=

0, it follows that ∂D lifts to a single separating loop in S, which exhibits S as a connectedsum of S ′ and S ′′. Hence H1(S) = H1(S

′) ⊕ H1(S′′) respecting the intersection form and

the Z/p action. Now (g(S ′), n(S ′)) = (0, n(S)) and (g(S ′′), n(S ′′)) = (g(S), 2), so knowingequation (5.1) for S ′ and S ′′ implies it for S. Hence it suffices to deal with the two casesg = 0 and n = 2.

Case g = 0. Embed a tree T = (V,E) in S, where V = {p1, . . . , pn−1} and where no edgepasses through pn. This gives a cell decomposition of S, which we lift to a (Z/p)-equivariantcell decomposition of S. Now let us consider the cellular chains computing H1(S). There isexactly one 2-cell, and it has zero boundary. Thus H1(S) is simply the group of 1-cycles.Now a 1-chain is fixed by Z/p iff for all e ∈ E it assigns the same weight to each of the plifts of e. Now it is easy to observe that since T is a tree, such a 1-chain cannot be a 1-cycleunless it vanishes. Thus H1(S)

Z/p = 0, so equation (5.1) holds in the case g = 0.Case n = 2. There is a natural exact sequence:

0 → H1(S,Z/p) → H1(S,Z/p)(r1,r2)−−−→ (Z/p)⊕2 +

−→ Z/p→ 0 (5.2)

Now, multiplying α by a nonzero element of Z/p yields an equivalent problem, so we assumewithout loss of generality that α ∈ A := (r1, r2)

−1(1,−1). We claim that MCG(S rel {p1, p2})acts transitively on A. Let [ℓ] ∈ A be the Poincare dual of a simple arc ℓ from p1 to p2. Weobserved earlier that MCG(S rel ℓ) acts transitively on the nonzero elements of H1(S,Z/p);hence MCG(S rel {p1, p2}) acts transitively on A \ {[ℓ]} (by exactness). On the other hand,since g ≥ 1, there certainly exists γ ∈ MCG(S rel {p1, p2}) for which γ[ℓ]− [ℓ] ∈ H1(S,Z/p)is nonzero (for instance, γ could be a Dehn twist around a nonseparating simple closedcurve which intersects ℓ exactly once), and so [ℓ] is in the same orbit as A \ {[ℓ]}. Thus

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MCG(S rel {p1, p2}) acts transitively on A. Hence we may assume without loss of general-ity that α = [ℓ]. Now we can see the cover S → S explicitly: we cut S along ℓ and gluetogether p copies in the relevant fashion. One then easily observes that there is an isomor-phism respecting intersection forms H1(S) ∼= H1(Σg)

⊕p, where Z/p acts by rotation of thesummands. Then H1(S)

Z/p is a copy of H1(Σg) with its intersection form multiplied by p.Hence equation (5.1) holds in the case n = 2 as well, and the proof is complete.

References

[1] Agol (mathoverflow.net/users/1345). Incompressible surfaces in an open subset ofRˆ3. MathOverflow. http://mathoverflow.net/questions/74935 (version: 2011-09-07).

[2] J. W. Alexander. On the subdivision of 3-space by a polyhedron. Proceedings of theNational Academy of Sciences of the United States of America, 10(1):pp. 6–8, 1924.

[3] R. H. Bing. An alternative proof that 3-manifolds can be triangulated. Ann. of Math.(2), 69:37–65, 1959.

[4] Salomon Bochner and Deane Montgomery. Locally compact groups of differentiabletransformations. Ann. of Math. (2), 47:639–653, 1946.

[5] Yu Qing Chen, Henry H. Glover, and Craig A. Jensen. Prime order subgroups ofmapping class groups. JP J. Geom. Topol., 11(2):87–99, 2011.

[6] Andreas Dress. Newman’s theorems on transformation groups. Topology, 8:203–207,1969.

[7] Michael Freedman, Joel Hass, and Peter Scott. Least area incompressible surfaces in3-manifolds. Invent. Math., 71(3):609–642, 1983.

[8] A. M. Gleason. The structure of locally compact groups. Duke Math. J., 18:85–104,1951.

[9] Andrew M. Gleason. Groups without small subgroups. Ann. of Math. (2), 56:193–212,1952.

[10] W. H. Gottschalk. Minimal sets: an introduction to topological dynamics. Bull. Amer.Math. Soc., 64:336–351, 1958.

[11] Robert D. Gulliver, II. Regularity of minimizing surfaces of prescribed mean curvature.Ann. of Math. (2), 97:275–305, 1973.

[12] A. J. S. Hamilton. The triangulation of 3-manifolds. Quart. J. Math. Oxford Ser. (2),27(105):63–70, 1976.

[13] William Jaco and J. Hyam Rubinstein. PL minimal surfaces in 3-manifolds. J. Differ-ential Geom., 27(3):493–524, 1988.

21

[14] Osamu Kakimizu. Finding disjoint incompressible spanning surfaces for a link. Hi-roshima Math. J., 22(2):225–236, 1992.

[15] Steven P. Kerckhoff. The Nielsen realization problem. Ann. of Math. (2), 117(2):235–265, 1983.

[16] Robion C. Kirby and Laurence C. Siebenmann. Foundational essays on topologicalmanifolds, smoothings, and triangulations. Princeton University Press, Princeton, N.J.,1977. With notes by John Milnor and Michael Atiyah, Annals of Mathematics Studies,No. 88.

[17] Joo Sung Lee. Totally disconnected groups, p-adic groups and the Hilbert-Smith con-jecture. Commun. Korean Math. Soc., 12(3):691–699, 1997.

[18] Iozhe Maleshich. The Hilbert-Smith conjecture for Holder actions. Uspekhi Mat. Nauk,52(2(314)):173–174, 1997.

[19] Gaven J. Martin. The Hilbert-Smith conjecture for quasiconformal actions. Electron.Res. Announc. Amer. Math. Soc., 5:66–70 (electronic), 1999.

[20] Mahan Mj. Pattern rigidity and the Hilbert–Smith conjecture. Geom. Topol.,16(2):1205–1246, 2012.

[21] Edwin E. Moise. Affine structures in 3-manifolds. V. The triangulation theorem andHauptvermutung. Ann. of Math. (2), 56:96–114, 1952.

[22] Deane Montgomery and Leo Zippin. Small subgroups of finite-dimensional groups. Ann.of Math. (2), 56:213–241, 1952.

[23] Deane Montgomery and Leo Zippin. Topological transformation groups. IntersciencePublishers, New York-London, 1955.

[24] M. H. A. Newman. A theorem on periodic transformations of spaces. Quart. J. Math.,os-2(1):1–8, 1931.

[25] J. Nielsen. Die struktur periodischer transformationen von flachen. Danske Vid. Selsk,Mat.-Fys. Medd., 15:1–77, 1937.

[26] Jakob Nielsen. Abbildungsklassen endlicher Ordnung. Acta Math., 75:23–115, 1943.

[27] Robert Osserman. A survey of minimal surfaces. Van Nostrand Reinhold Co., NewYork, 1969.

[28] Robert Osserman. A proof of the regularity everywhere of the classical solution toPlateau’s problem. Ann. of Math. (2), 91:550–569, 1970.

[29] C. D. Papakyriakopoulos. On Dehn’s lemma and the asphericity of knots. Ann. ofMath. (2), 66:1–26, 1957.

22

[30] Piotr Przytycki and Jennifer Schultens. Contractibility of the Kakimizu complex andsymmetric Seifert surfaces. Trans. Amer. Math. Soc., 364(3):1489–1508, 2012.

[31] Frank Raymond and R. F. Williams. Examples of p-adic transformation groups. Ann.of Math. (2), 78:92–106, 1963.

[32] Dusan Repovs and Evgenij Scepin. A proof of the Hilbert-Smith conjecture for actionsby Lipschitz maps. Math. Ann., 308(2):361–364, 1997.

[33] J. Sacks and K. Uhlenbeck. The existence of minimal immersions of 2-spheres. Ann. ofMath. (2), 113(1):1–24, 1981.

[34] J. Sacks and K. Uhlenbeck. Minimal immersions of closed Riemann surfaces. Trans.Amer. Math. Soc., 271(2):639–652, 1982.

[35] R. Schoen and Shing Tung Yau. Existence of incompressible minimal surfaces and thetopology of three-dimensional manifolds with nonnegative scalar curvature. Ann. ofMath. (2), 110(1):127–142, 1979.

[36] Jennifer Schultens. The Kakimizu complex is simply connected. J. Topol., 3(4):883–900,2010. With an appendix by Michael Kapovich.

[37] P. A. Smith. Transformations of finite period. III. Newman’s theorem. Ann. of Math.(2), 42:446–458, 1941.

[38] Edwin H. Spanier. Cohomology theory for general spaces. Ann. of Math. (2), 49:407–427, 1948.

[39] Norman E. Steenrod. Universal Homology Groups. Amer. J. Math., 58(4):661–701,1936.

[40] Peter Symonds. The cohomology representation of an action of Cp on a surface. Trans.Amer. Math. Soc., 306(1):389–400, 1988.

[41] Terence Tao. Hilbert’s fifth problem and related topics. Manuscript, 2012.http://terrytao.wordpress.com/books/hilberts-fifth-problem-and-related-topics/.

[42] William P. Thurston. On the geometry and dynamics of diffeomorphisms of surfaces.Bull. Amer. Math. Soc. (N.S.), 19(2):417–431, 1988.

[43] Friedhelm Waldhausen. On irreducible 3-manifolds which are sufficiently large. Ann. ofMath. (2), 87:56–88, 1968.

[44] John J. Walsh. Light open and open mappings on manifolds. II. Trans. Amer. Math.Soc., 217:271–284, 1976.

[45] David Wilson. Open mappings on manifolds and a counterexample to the Whyburnconjecture. Duke Math. J., 40:705–716, 1973.

23

[46] Hidehiko Yamabe. On the conjecture of Iwasawa and Gleason. Ann. of Math. (2),58:48–54, 1953.

[47] Hidehiko Yamabe. A generalization of a theorem of Gleason. Ann. of Math. (2), 58:351–365, 1953.

[48] Chung-Tao Yang. p-adic transformation groups. Michigan Math. J., 7:201–218, 1960.

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